An early goal from Borna Barisic opened the scoring at Ibrox before Fabio Silva found the back of the net in the second-half to see Philippe Clement’s men into the last eight of the competition.
There were seven changes from the Scottish Premiership victory over Aberdeen including first starts for January signings Mohamed Diomande and Oscar Cortes. Jack Butland retained his place in goal behind a back four of James Tavernier, Connor Goldson, John Souttar and Borna Barisic.
Diomande came into midfield alongside Nico Raskin and Ryan Jack, while Cortes and Scott Wright supported striker Fabio Silva.
Ben Davies was in the matchday squad for the first time since December as he returns from injury and was named on the bench alongside Robby McCrorie, Ridvan, John Lundstram, Cyriel Dessers, Todd Cantwell, Rabbi Matondo, Leon King and Ross McCausland.
With a place in the quarter-finals up for grabs, Rangers looked to put their Championship opponents under pressure from the whistle with Cortes almost finding the bottom corner with a low curling strike but it came off the outside of the post.
The winger had another chance shortly after, but his right-footed strike was held by Robbie Mutch in the Ayr United goal.
It was 1-0 with less than 10 minutes on the clock as the early pressure from Rangers paid off. The move started with John Souttar who found Scott Wright just inside the Ayr United half and he picked Oscar Cortes driving down the left.
The Colombian’s initial cross was blocked by an Ayr defender, but he was alert to win the ball back again before it fell for Borna Barisic who was in a good position inside the box to release a left-footed strike towards goal which took a slight deflection on its way beyond Robbie Mutch.
Ayr United were awarded a corner as Mark McKenzie broke down the right with his dangerous cross headed behind by Ryan Jack, but the set-piece was dealt with by Rangers.
Rangers were having joy down the left, and again created another opportunity with Fabio Silva flicking a header just wide from a Borna Barisic delivery.
Barisic almost fed in Silva again with a chip over the Ayr defence but the visitors stood strong to deny the on-loan Wolves striker an effort at goal.
On the stroke of half-time, Rangers almost had a second as Scott Wright found Borna Barisic in the box with a good delivery with the full-back’s shot deflected towards goal by George Stanger, but Mutch got a block to keep the ball out.
An Ayr corner early in the second-half was struck towards goal by Ben Dempsey who had found space in the box but it was off target from the midfielder. At the other end Ryan Jack drilled a low shot from the edge of the box just wide as he got on the end of a Scott Wright delivery.
There were two goalline clearances from the visitors as Diomande and Cortes both saw efforts blocked by the Ayr defence after Wright weaved his way into the box, but the Championship side held strong.
With 60 minutes played at Ibrox, Philippe Clement turned to his bench for the first time this evening as Rabbi Matondo, Todd Cantwell and Ridvan were introduced in place of Oscar Cortes, Mohamed Diomande and Borna Barisic. For Matondo, it was his 50th competitive appearance in Light Blue.
Fabio Silva almost got on the end of a James Tavernier free-kick as Rangers continued to search for the second goal at Ibrox.
John Lundstram was next to be introduced by Clement as he replaced Ryan Jack for the final 20 minutes.
That all important second goal did come for Rangers with 76 minutes on the clock at Ibrox. Cantwell’s chip towards goal was cleared on the line by an Ayr defender before Silva released a powerful strike high into the back of the net for his second goal for the club.
A final change for Rangers saw Leon King introduced in place of Nico Raskin for the final six minutes of the cup tie.
Matondo had a low strike blocked as Gers continued to push for a third, as the fourth official indicated five minutes of added time.
Rangers will discover their quarter-final opponents tomorrow evening, with the draw live on BBC One Scotland ahead of Airdrieonians v Hearts.
RANGERS: Butland, Tavernier, Goldson, Souttar, Barisic (Ridvan, 60), Raskin (King, 84), Jack (Lundstram, 71), Diamonde (Cantwell, 60), Cortes (Matondo, 60), Wright, Silva.
Subs: McCrorie, Dessers, Davies, McCausland.
Ayr United: Mutch, McAllister (Tomlinson, 63), Stanger, McGinty, Reading, Syla, Dempsey, Murphy (Guthrie, 82), Chalmers (Bryden, 63), McKenzie (Willoughby, 78) Dowds.
Subs: Albinson, Musonda, McGeady, McHugh, Sanders.